. :
11 2+ 22 2 + a 33 z 2 +2 12 + 2 13 + 2 23 z +
+ 2 1 + 2 2 +2 a 3 z+ c = 0, (8)
aij, i, j =1,2,3 . , , .
(8) . Ÿ
a 11 12 13
A = 12 22 23
13 23 33
,
λ1 0 0
A ¢ = 0 λ2 0
00λ3
Ox ¢ y ¢ z ¢ (8) :
λ1 x ¢2+ λ2 y ¢2 + λ3 z ¢2, (9)
. λ1, λ2, λ3 Ox ¢ y ¢ z ¢. .. (9), λ1, λ2, λ3 ( ). λ i , , . , λ i, , . , λ i , λ3. λ i ≠ 0, λ1.
λ1, λ2, λ3 : (+, +, +), (+, +, ), (+, +, 0), (+, , 0), (+, 0, 0). . :
λ1 x ¢2+λ2 y ¢2+λ3 z ¢2 +2 b 1 x ¢+2 b 2 y ¢+2 b 3 z ¢ + c = 0(10)
, λ i ≠ 0
λ1(x ¢2 + x ¢+) + λ2(y ¢2 + y ¢+) +
+ λ3(z ¢2 + z ¢+) + c = 0,
λ12+ λ22+ 2+ c ¢= 0.
.
x ² = x ¢+, y ² = y ¢+, z ² = z ¢+,
O ¢(,, ).
λ1(x ²)2+λ2(y ²)2 + λ3(z ²)2 = '.
| ¢|, ¢ ≠ 0. 1, 1 0. 1 6 (. ).
λ3= 0, z ¢, 2 b 3 z ¢ + c ¢ : 2 b 3(z ¢ + c ¢/2 b 3), z ² = z ¢+ c ¢/2 b 3. 2 b 3 z'', , (z ²)2:
|
|
λ1(x ²)2+λ2(y ²)2 = 2 b 3 z''.
| b 3| 7, 8 (. ). 2 z'', 2 z'', , - .
λ3= 0 b 3= 0,
λ1(x ²)2 +λ2(y ²)2 = ',
9 13 .
λ2= λ3= 0. , , (8) :
E | ||
1. | + + =1, | d>0 ∆<0 |
2. (Ø) | + + = 1, | d>0 ∆>0 |
3. () | + + = 0, | d>0 ∆=0 |
4. | + = 1, | d<0 ∆<0 |
5. | + = 1, | d<0 ∆>0 |
6. | + = 0, | d<0 ∆=0 |
7. | + = 2 z, | d=0 ∆<0 |
8. | = 2 z, | d=0 ∆>0 |
9. | + =1, | d=∆=0 I2>0 I1∙I4<0 |
10. | + = 1, | d=∆=0 I2>0 I1∙I4>0 |
11. , | + = 0, | d=∆=0 I2>0 I4=0 |
12. | = 1, | d=∆=0 I2<0 I4≠0 |
13. | = 0, | d=∆=0 I2 <0 I4=0 |
14. | x 2 = 2 py | d=∆=0 I2=0 I4≠0 |
15. | x 2 = a 2 | d=∆=0 I2=I4=0 I3>0 |
16. | x 2 = 0 | d=∆=0 I2=I4=0 I3=0 |
17. | x 2 = a 2 | d=∆=0 I2=I4=0 I3<0 |
:
11 12 13 a 11 a 12 a 13 a 1
d = 12 22 23 a 12 a 22 a 23 a 2
13 23 a 33 a 13 a 23 a 33 a 3
a 1 a 2 a 3 c
I1 = trace d = a 11 + a 22 + a 33
I2 = + + -
d
-
I3 = + + ∆,
d.
a 11 a 12 a 11 a 11 a 13 a 1 a 22 a 23 a 2 -
I4 = a 12 a 22 a 2 + a 13 a 33 a 3 + a 23 a 33 a 3
a 1 a 2 c a 1 a 3 c a 2 a 3 c
∆, d
. δ, ∆, I1, I2 , I3, I4 , ( ).
. , . .
1. . :
|
|
4 x 2 + z 2 24 x + 8 y + 2 z + 5 = 0.
. :
4(x 2 6 x + 9) 36 + (z 2 + 2 z +1) - 1 + 8 y + 5 = 0,
4(x 3)2 + (z + 1)2 + 8 y 32 = 0,
4(x 3)2 + (z + 1)2 = 8(y 4).
:
x ¢ = x 3,
y ¢= y 4,
z ¢ = z + 1,
O' (3, 4,1).
4 x ¢2 + z ¢ 2 = 8 y ¢, Û(x ¢)2 + = 8 y ¢,
, . + = 1 4 8. .
2. ,
(x 3) 4(y + 4)2 ³ 0.
.
. , (x 3) 4(y + 4)2 = 0.
x ¢ = x 3,
y ¢= y + 4,
z ¢ = z .
O' (3, 4, 0). ,
(x ¢ )2 4(y ¢ )2 = 0. (*)
z ¢, , O ¢ z ¢. O ¢ x ¢ y ¢ (*)
(x ¢ 2 y ¢)(x ¢ + 2 y ¢ ) = 0.
2 . , . , . , , . O ¢ x ¢: A (a, 0, 0) O ¢ x ¢ y ¢ z ¢ , (x ¢ )2 4(y ¢ )2 ³ 0. , O ¢ x ¢ . , , . .. , .
3. ,
z= 2 y 2,
x= 0.
a) Oz,) Oy . . .
. l, Oyz. l . , Φ, l Oz, l z , y : y a.
z= 2 2, Þ
x 2 + y 2 = 0.
, Oz, O ¢(0, 0,2).
.
1) z = 2, x 2 + y 2 = 4. , z = 2 2. z = 2 Oz , Ox Oy; 2 , . , Ox , Oy Oz.
2) z = 6 Oz.
3) O ¢. , .
4) , .
, Oy, l y , z : z a.
|
|
x 2 4(y 1)2 + z 2 = 0,
(y 1)2 + = 0.
, O ¢(0, 1, 0), Oy. . , , y = 3 4.
4.
x 2 + + = 1
? , ( ) ? .
. . :
x 2 + = 1,
x z . , Oy. , , g , y x 2 + = 1 , z = 0. , ,
x 2 + = 1
z = 0
z, γ', y = 0:
x 2 + = 1,
y =0.
5. , x = a F (a, 0, 0).
. M (x, y) .
| MF | =,
M | x + a |.
= | x + a |.
:
x 2 2 ax + a 2 + y 2+ z 2 = x 2+ 2 ax + a 2
y 2+ z 2 = 4 ax Û + = 2 ax.
, Ox.
6. + + = 1 = =.
. :
x = 1 + t,
y = 4 6 t,
z = 6 + 12 t.
:
+ + = 1,
1 + 2 t + t 2 + 4 12 t + 9 t 2 + 4 16 t + 16 t 2 = 9,
26 t 2 26 t = 0 Û 26 t (t 1) = 0.
: t 1= 0, t 2= 1. , M 1(1, 4, 6), M 2(2, 2, 6).
: M 1(1, 4, 6), M 2(2, 2, 6).
7. , + = 1 ) y = 2 z; ) y = 2 z +2.
. ) . y = 2 z :
+ = 1 Û x 2 = 9.
Oxz. . , .
) y = 2 z +4 :
+ = 1 Û + = 1 Û x 2 = 9 z.
Oxz. . , .
.
, . , , ( ) , ( ) , . ,
1 2 3 4
5 6 7 8
, 2 4 . , 2´4. a 11= 1, a 12= 2, a 21= 6. n ´ n n.
, , . , , , . , E. , 3
|
|
1 0 0
E = 0 1 0
0 0 1
, () , ().
. A det A. , . 2 :
a 11 a 12
a 21 a 22
Mij , A i - j - . , aij. 3 :
a 11 a 12 a 13
a 21 a 22 a 23 = a 11 M 11 a 12 M 12 + a 13 M 13 = a 11 a 12 + a 13.
a 31 a 32 a 33
.
1 2 3
4 5 6 =1 2 + 3 = 1 (5 9 6 8) 2 (4 9 67) + 3 (48 57) =
7 8 9
= 3 +12 9 = 0.
.
1. , .
2. ( ), .
3. () .
4. () .
. 3 :
1 2 3 1 2 1
4 5 6 = 3 4 5 2
7 8 9 7 8 3
5. () (), , .
( ):
1 2 3 1 2 3
4 5 6 = 3 3 3
7 8 9 6 6 6
, , .
6. :
1 2 1
0 3 6 = 1 (3) 9 = 27
0 0 9
. .
.
, . , 3:
a 11 x 1 + a 12 x 2+ a 13 x 3 = b 1
a 21 x 1+ a 22 x 2 + a 23 x 3 = b 2 (1)
a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3.
aij , b 1, b 2, b 3 . A, B:
a 11 a 12 a 13 b 1
A = a 21 a 22 a 23 B = b 2
a 31 a 32 a 33 b 3.
D = det A, D i , A i - B. ,
b 1 a 12 a 13
D1= b 2 a 22 a 23
b 3 a 32 a 33.
. ( ). D¹0, (1) .
x 1 =, x 2 =, x 3 =.
, n .
.
5 x + 9 y = 3,
3 x + 5 y = 1.
.
5 93 9 5 3
3 5 1 5 3 1
x 1 = = = 3, x 2 = = = 2.
: (3, 2).
90
17
20
16, 19
18, 20
16, 19
15, 19
6
, 7
48
7
8
12
22
90
123
118
94
87
121
62
51
62
89
91
92
117, 132
125, 132
125, 132
28
17
20
21
97
87
92
2 101
2 132
|
|
90
92
48, 64
87
94
100, 130
7
7
118
92
16, 17, 19
16, 17, 19
100
12
100, 130
116
140
102
123,
125, 128
104
6
18, 21
48, 63
116, 118
17
56
116, 118
2 97
127
60
50
953
140
11
54
17, 20
17, 20
12
117, 121, 132
105
105
11
11
94
117, 132
128
128
121
45, 47
92
48, 64
88
31
121
130
32
90
87
29
29
29
7
8
12
138
31
34
10, 21
12
12
9
7
6
57
() 58
() 58
89
- 17, 19
17, 19
9
11
68
54
16, 17
19, 20
100, 130
131
12
14
14
26
7
6
98
90
98
7
30
30
122
12
11
51
11
62
52, 53, 67
44, 46
45, 46
59
61
54
48
54
51
86, 89, 92
91
104
97
101
116
30
30
6
93
86
123, 132
117, 132
1. .. . .:,1978
2. .. . .: , 1984.
3. .., .. . .I. .:, 1986.
4. .., .. . . .I. .: , 1974
5. .. . .I. .: , 1973
6. .., .. . .: , 1981.
7. .M. . .:, 1973.
8. .M. . I: . .:, 1986.
9. .., .. . .: , 1990.
10. .., .. . : ,1989
11. .. . . .:, 1980.
12. .., .. . .1. .: , 1973.
13. . . .. : - , 1989, 1999.
14. .. . .: , 1987
15. .. . .:, 1965
16. .., .. . .: , 1986
17. .., .. . .: , 1976
18. .. . .: - , 1969